In this section, methods for inference with interference in observational studies are re- viewed and a new approach is proposed. Let Yij(zi) be the binary outcome (e.g., malaria)
for individual j = 1, . . . , ni in group i = 1, . . . , k when group i has exposure zi (e.g., bed
net use). Let zi ∈ Z(ni) where Z(ni) contains the 2ni realizations of Zi, e.g., Z(2) =
{(0,0),(0,1),(1,0),(1,1)}. The vector of exposures can be partitioned as zi = (zij, zi(j)) wherezij is the treatment assignment for individual j and zi(j) is the treatment assignment for all individuals except individual j. In this formulation, both Yij(zi) and Zi are random
variables. Throughout, partial interference (Sobel, 2006) is assumed such that interference can only occur within groups. Thus, each individual’s potential outcomes are a set of random variables{Yij(zi) :zi ∈Z(ni)}. Suppose that each individual independently selects treatment
with probabilityα. In this scenario, each individual’s set of potential outcomes can be sum- marized by weighted averages. Let πα(ω|z) = Prα(zi(j) =ω|zij =z). The average potential
outcome for individualj in group igiven zij =z equals
¯ Yij(z;α) = X ω∈Z(ni−1) πα(ω|z)Yij(zij =z;zi(j)=ω) = X ω∈Z(ni−1) α|ω|(1−α)ni−1−|ω|Y ij(zij =z;zi(j)=ω)
in which zi(j) ∈Z(ni−1) is the treatment vector of all individuals in group i except j, and
|ω| denotes the sum of ω. The group i average potential outcome on treatment z = 0,1 under treatment strategy α equals ¯Yi(z;α) = n−i 1
Pni
distribution function of ¯Yi(z;α), and define
µz,α ≡E[ ¯Yi(z;α)] =
Z 1
0
ydFz,α(y)
Letπα(x) = Prα(zi=x). Assuming that each individual independently selects treatment
with probabilityα, the average potential outcome for individualj in groupiunderα is equal to ¯ Yij(α) = X ω∈Z(ni) πα(ω)Yij(zi=ω) = X ω∈Z(ni) α|ω|(1−α)ni−|ω|Y ij(zi =ω) = X z=0,1 X ω0∈Z(n i−1) αz(1−α)1−zα|ω0|(1−α)ni−1−|ω0|Y ij(zij =z;zi(j)=ω0) = (1−α) ¯Yij(0;α) +αY¯ij(1;α) (4.9)
The group i average potential outcome under treatment strategy α is equal to ¯Yi(α) =
n−i1Pni
j=1Y¯ij(α). Finally, let µα≡E[ ¯Yi(α)] = (1−α)µ0,α+αµ1,α.
Extending Hudgens and Halloran (2008) to the superpopulation setting, the inferential targets are
DE(α) =µ0,α−µ1,α
IE(α, α0) =µ0,α−µ0,α0
T E(α, α0) =µ0,α−µ1,α0
OE(α, α0) =µα−µα0 (4.10)
The observed data reveal for groupi= 1, . . . , kthe treatment vectorZi = (Zi1, . . . , Zini),
4.4.1 Inverse Probability Weighted Estimation
Tchetgen Tchetgen and VanderWeele (2012) showed that
ˆ YiIP W(z;α) = 1 ni ni X j=1 πα(Zi(j)|Zij)1{Zij =z}Yij Pr[Zi|Xi] (4.11) and ˆ YiIP W(α) = 1 ni ni X j=1 πα(Zi)Yij Pr[Zi|Xi] (4.12)
are unbiased for ¯Yi(z;α) and ¯Yi(α), respectively, in a finite population setup assuming (4.3)
when the propensity score Pr[Zi|Xi] is known. The estimators (4.11) and (4.12) are also
unbiased in the superpopulation setting, as in the following propositions:
Proposition 4.1. E[ ˆYiIP W(z;α)] =µz,α Proposition 4.2. E[ ˆYiIP W(α)] =µα
Proofs of Proposition 4.1 and 4.2 are given in the Appendix. In observational studies, Pr[Zi = z|Xi] is rarely known and must be estimated, e.g., by a regression model. The
product of individual level model predicted probabilities serves as the estimator cPr[Zi|Xi]. When the propensity score is known or estimated correctly, Perez-Heydrich et al. (2014, Web appendix) use M-estimation theory to show that the IPW estimators in (4.4) are consistent for their target parameters and asymptotically normal. Additionally, they present sandwich variance estimators to be used in confidence intervals.
4.4.2 Outcome Modeling
As noted previously, under partial interference, individual j in group i has 2ni potential
outcomes. Asni increases, the computational difficulties associated with this problem mount
considerably. In many settings, it may be reasonable to consider functional assumptions about interference
Under (4.13), an individual’s potential outcome on treatmentz= 0,1 is the same when some function of the otherni−1 treatment assignments maps to the same value. At one extreme,
when f(x) = x, each individual has 2ni potential outcomes, and at the other extreme when
f(x) = 0, each individual only has two potential outcomes (no interference). One reasonable function is f(x) = 1{|x| > 0}, where |x| denotes the sum of the elements in x. Under this threshold function, each individual has four potential outcomes, {yij(zij =z,1{|zi(j)|>0}= c)} forz, c = 0,1. Another reasonable assumption is stratified interference, or the function f(x) =|x|. Under this assumption, each individual has 2ni potential outcomes. The purpose
of the functional assumption (4.13) is to simplify the problem, mapping the 2ni group vector
assignments to the real line. Under (4.13), forz= 0,1
¯ Yij(z;α) =Pc∈C n P ω:f(ω)=cπα(zi(j)=ω|zij =z) o Yij(zij =z;f(zi(j)) =c)
A statistical model for E[Yij(zij = z;f(zi(j)) = c)] can be used for inference on the causal effects (4.10), e.g.,
L(Pr[Yij(zij =z;f(zi(j)) =c) = 1]) =β0+β1z+β2c+ηXi (4.14)
in which Pr[Yij(zij = z;f(zi(j)) = c) = 1] = E[Yij(zij = z;f(zi(j)) = c)] as Y is binary. The parameter estimates for model (4.14), the resulting causal effect estimators, and the corresponding large sample distributions can be derived also using M-estimation theory.
4.4.3 A Bayesian Approach
A Bayesian approach to evaluating causal exposure-outcome relationships in observational studies with interference is outlined in this section. Although similar to the outcome modeling approach, the Bayesian approach has the key advantage of accommodating missing covariate data in a straightforward manner, an important feature given that the DHS contain missing covariate data. Using model (4.14) for the potential outcomes, letθ= (β0, β1, β2, η). Inference on the causal effects is carried out by sampling from their posterior distributions using an analogous Gibbs sampler to the one outlined in Section 2. In this Gibbs sampler, each
individual will have greater than or equal to two potential outcomes, so thatYmis will be of length greater than or equal to one for each individual. The causal effects (4.10) are functions ofθ and can be directly computed in step 3 of the Gibbs sampler.
4.4.4 A Simulation Study
To study the proposed methods, the naive, IPW, outcome, and Bayesian point and in- terval estimators were compared in a simulation study motivated by the DHS that assumed stratified interference. As will become clear in the simulation below, the true causal effects were functions ofµz,α= EX,ni 1 ni ni X j=1 ni−1 X c=0 ni−1 c αc(1−α)ni−1−cL−1 β0+β1z+β2 z+c ni +β3Xij +β4z z+c ni (4.15) and µα = (1−α)µ0,α+αµ1,α. The derivation of (4.15) is given in the appendix.
The simulation proceeded as follows:
1. For groupi= 1, . . . , k,ni was randomly sampled fromf(ni)
2. For individualj in groupi, a Xij was randomly sampled fromf(X)
3. For individualjin groupi, bed net statusZij was self-selected using the following model
Zij ∼Bernoulli(L−1(γ0+γ1Xij)) (4.16)
4. Given Zi = (Zij, Zi(j)) from step 3, an observed malaria outcome for individual j in groupiwas generated using the model
Yij(zij =z, zi(j)=ω)∼Bernoulli(L−1(β0+β1z+β2(|ω|+z)/ni+β3Xij+β4z(|ω|+z)/ni))
(4.17)
5. To mimic a two-stage randomized experiment, for groups where P
jZij/ni ∈ [αs −
were excluded and the estimators of Hudgens and Halloran (2008) and Wald-type 95% confidence intervals of Liu and Hudgens (2014) were computed as the naive estimators.
6. IPW estimators and 95% confidence intervals were computed wherein the propensity scores were treated as known.
7. IPW estimators and 95% confidence intervals were computed wherein the propensity scores were estimated using the correct model (4.16).
8. Outcome frequentist estimators and 95% confidence intervals were computed using the correct model (4.17).
9. Bayesian posterior means and 95% credible intervals were computed using the correct model (4.17) and the priorsβm∼Normal(0,4) for m= 0, . . . ,3 with a burn-in of 1100,
a thinning interval of 3, and 1000 samples for the posterior distribution.
10. Steps 1-9 were repeated 1000 times.
The simulations with interference were also motivated by the DHS such that for individual jin groupi,Yij(zi) was the binary potential malaria outcome,Zij was the binary exposure of
bed net use,P
jZij/ni was the proportion of bed net use in groupi, andXij was the binary
confounder of proximity to an urban space. The distributions ofniandXwere estimated from
the DHS such thatni ranged from 4-52 with a mean of 29.48 and such that Pr[X = 1] = 0.45.
In computing the naive estimator, w = 0.05. The parameters in models (4.16) and (4.17) were estimated using the DHS such that γ0 = −1.61, γ1 = 0.47, β0 = −0.47, β1 = −0.40, β2 = −0.13, β3 = −0.63, β4 = −0.06. Additional simulation inputs are summarized in Figure 4.1.
Simulation results are given in Figure 4.2 forα= 0,0.1, . . . ,1. The IPW estimators can be biased, can have ESE/ASE ratios much larger than 1, can fail to achieve nominal coverage, and can have much larger widths than outcome models whereas the outcome models are unbiased, have ESE/ASE ratios near 1, achieve nominal coverage, and have sensible widths. The overall performance of the IPW estimators was noticeably worse forαvalues in low probability areas
of the observed range of group coverage, i.e.,α≤0.1 orα≥0.4. For example, IPW estimators forα= 0.2 had good operating characteristics whereas IPW estimators forα= 0.8 had poor operating characteristics.